Sr₂RuO₄ is an unconventional superconductor distinguished by its potential chiral *p*-wave order parameter. This chiral nature, specifically the *k*x ± *i*ky form, is supported by various experimental observations like muon spin relaxation and high-resolution polar Kerr effect measurements. Recent nuclear magnetic resonance experiments further suggest the *d*-vector is not parallel to the *c*-axis, potentially indicating chiral *d*-wave states. The chiral state's double degeneracy allows for superconducting domains of different chirality and a spontaneous edge current, a feature yet to be directly observed. Although indirect evidence has been found through transport experiments utilizing Ru inclusions to create proximity junctions, understanding of Sr₂RuO₄ is hampered by the lack of direct observation of these domains or edge currents. A complicating factor is the existence of a "3-K phase", possibly a non-chiral state with a single-component order parameter, induced by symmetry breaking due to Ru inclusions or uniaxial strain. This paper focuses on the "intrinsic phase" with *T*c ≈ 1.5 K, associated with the pure bulk material, differentiating it from the extrinsic phase with *T*c ≈ 3 K. Most prior research has been limited to bulk crystals, hindering the study of chiral domains, which are anticipated to be only a few microns in size. The mobility of these domains, suggested by time-dependent switching noise in transport measurements, adds another layer of complexity. Mesoscopic samples offer a promising platform to investigate and potentially control these domains, as the energy cost associated with a chiral domain wall (ChDW) increases with area. Recent theoretical work suggests that mesoscopic chiral *p*-wave superconductors could exhibit multichiral states, with ChDWs acting as Josephson junctions due to the local suppression of the order parameter. This study utilizes mesoscopic rings of Sr₂RuO₄ to investigate this hypothesis.

The literature extensively discusses the unusual superconducting properties of Sr₂RuO₄, particularly the possibility of a chiral p-wave pairing state. Early evidence from muon spin relaxation and the polar Kerr effect pointed towards broken time-reversal symmetry, a hallmark of chiral superconductivity. However, the precise nature of the superconducting order parameter remained elusive, with debates surrounding the role of multiple components and potential chiral d-wave states. Previous transport measurements, often using Ru inclusions to form artificial junctions, suggested the existence of dynamical superconducting order parameter domains and anomalous switching behavior, hinting at the presence of chiral domain walls. The study of mesoscopic structures, as proposed by Fernández Becerra and Milošević, provides a new avenue to investigate these multichiral states and the behavior of chiral domain walls acting as intrinsic Josephson junctions. The Little-Parks effect in superconducting rings, stemming from flux quantization, offers a complementary method to probe the superconducting state, providing a contrasting behavior to the expected Josephson junction oscillations.

Single crystals of Sr₂RuO₄ were grown using the floating zone method and subsequently structured into microrings via focused ion beam (FIB) milling. Two types of rings (Rings A and B) were fabricated with slightly different inner and outer radii, both exhibiting thicknesses around 0.7 µm. The temperature-dependent resistance R(T) of the rings was measured, demonstrating sharp superconducting transitions consistent with bulk Sr₂RuO₄ and high residual resistivity ratios, indicating high-quality samples. FIB milling's effect on the intrinsic properties of Sr₂RuO₄ was verified by comparing the R(T) curves before and after milling. Current-voltage (V(I)) characteristics were also measured, showing negligible hysteresis, further confirming the high quality of the samples. Time-dependent Ginzburg-Landau (TDGL) simulations were employed to predict the chiral-domain configurations within the microrings under varying conditions, including the absence and presence of a magnetic field and different temperatures. These simulations modeled the Cooper-pair density and identified the energetically favorable domain configurations as a function of temperature and ring dimensions, relative to the coherence length ξ(T). The simulations predicted the formation of ChDWs, acting as weak links, even in the absence of a magnetic field. Supercurrent interference in the rings was investigated by measuring the critical current (Ic) as a function of magnetic field (H). The measurements were performed at various temperatures, both deep within the superconducting state and close to Tc. Magnetoresistance measurements, obtained by applying a constant DC current while sweeping the magnetic field, were also conducted to further analyze the supercurrent interference. These measurements provided insights into the behavior of the Josephson junctions at different temperatures and magnetic fields. For comparison, a ring (Ring C) with a broader superconducting transition (indicative of the extrinsic phase) was also studied. Its magnetoresistance was measured and compared to the results from Rings A and B. The Little-Parks oscillations, resulting from flux quantization in the ring, were simulated and compared to experimental data to distinguish the oscillations from the Josephson junction behavior.

Rings A and B, characterized by sharp superconducting transitions around 1.5 K (intrinsic phase), displayed distinct critical current (Ic) oscillations as a function of magnetic field. The period of these oscillations corresponded precisely to the flux quantum, characteristic of a DC SQUID with a pair of parallel Josephson junctions. Crucially, these oscillations persisted over the entire temperature range below Tc, with consistent overall shape despite changes in the coherence length, indicating that the junctions are intrinsic to the material. In contrast, rings with a broader transition (around 3 K, extrinsic phase), such as Ring C, showed no Ic oscillations. Instead, they exhibited standard Little-Parks oscillations, consistent with variations in Tc due to flux quantization. The amplitude of the oscillations in Rings A and B significantly exceeded what could be attributed solely to Tc variations, further supporting the presence of intrinsic Josephson junctions. TDGL simulations confirmed the theoretical possibility of spontaneous ChDW formation in the mesoscopic rings, explaining the observed behavior. The simulations predicted the emergence of two parallel Josephson junctions formed by ChDWs, consistent with the experimentally observed Ic oscillations. The simulations also showed that the Meissner state (without ChDWs) can be stabilized by increasing the ring arm width relative to the coherence length. The study clearly differentiated the Little-Parks oscillations, driven by Tc variations, from the larger amplitude magnetoresistance oscillations driven by Ic(H) oscillations, providing conclusive evidence supporting the existence of intrinsic Josephson junctions in the 1.5K phase of Sr2RuO4. Several alternative mechanisms for the observed oscillations were ruled out. These include circulating persistent currents, current-excited moving vortices, geometric constrictions as Josephson junctions, and accidental proximity or tunnel junctions. The absence of hysteresis in the V(I) curves, along with the consistent shape of the interference patterns over a wide temperature range, strongly supported the ChDW scenario.

The observed critical current oscillations in the mesoscopic Sr₂RuO₄ rings provide compelling evidence for the existence of chiral domain walls (ChDWs) acting as intrinsic Josephson junctions in the intrinsic (1.5 K) phase. The findings strongly support the chiral nature of the superconducting state in this material. The absence of these oscillations in rings exhibiting the extrinsic (3 K) phase suggests a direct correlation between the chiral superconductivity and the formation of the junctions. The results significantly advance our understanding of the superconducting order parameter in Sr₂RuO₄, addressing the long-standing puzzle of its complex domain structure. The high-quality of the samples, confirmed by the negligible hysteresis in the V(I) characteristics and the high residual resistivity ratios, ensures the robustness and reliability of the findings. The agreement between experimental observations and TDGL simulations further strengthens the interpretation of the results.

This study provides compelling evidence for the existence of intrinsic Josephson junctions in mesoscopic rings of Sr₂RuO₄, arising from spontaneously formed chiral domain walls. The precise nature of the degenerate states is not fully resolved and requires further investigation. The work highlights the power of combining theoretical simulations with mesoscopic experiments to study superconducting domains in unconventional superconductors and opens up new avenues for exploring topological quantum phenomena. Future research could focus on clarifying the exact nature of the domain walls, exploring the transport characteristics of these junctions in more detail, and investigating potential applications in topological quantum computing.

The study focuses on mesoscopic rings with specific dimensions and fabrication methods. The generalizability of the findings to other geometries and fabrication techniques requires further investigation. While alternative mechanisms for the observed oscillations were ruled out, the possibility of subtle contributions from other effects cannot be entirely excluded. Furthermore, the simulations rely on the Ginzburg-Landau formalism, which is an approximate model, and may not fully capture the complexities of the real material.